Let \(\epsilon > 0\)  be a small real number and consider the map
\(\Phi ^{\scriptscriptstyle (}k\) )\(\)  [0, 1 - ] Aut(Sk(F2))\( which satisfies the differential equation\begin{equation}{d\,\Phi ^{\scriptscriptstyle (}k\scriptscriptstyle )}\end{equation}_{\epsilon }(x)\over dx\)  = k 1 i 5 A\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) i x - i \(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) (x)             \(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) (0) = Id\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) .

The limit of this map as \(\epsilon \rightarrow 0\)  will play a central role in proving Theorem REF .
Differential equations such as () were first considered by Poincaré who gave explicit solutions of such equations
[1]}. Lappo-Danilevsky, among many other results, computed the monodromy of these solutions, see mémoire second in [2]}. The solutions are given in terms of hyperlogarithms which, for the specific case of (), are defined upon the choice of a tuple \((\zeta ^{i_1}, \dots , \zeta ^{i_r}) \in (\mu _5)^r\)  by the iterated integral
\(L_0(\zeta ^{i_1}, \dots , \zeta ^{i_r} \vert 1-\epsilon ):= \int _0^{1-\epsilon }\hspace{-3.33328pt}{1\over s_{r} - \zeta ^{i_r}} \int _0^{s_{r}}\hspace{-3.33328pt}{1\over s_{{r-1}} - \zeta ^{{i}_{r-1}}} \cdots \int _0^{s_{2}}\hspace{-3.33328pt}{1\over s_{1} - \zeta ^{{i}_1}} ds_{1}\cdots ds_{{r-1}} ds_{{r}}.\) 
